ON THE EXISTENCE OF HOMOCLINIC SOLUTIONS FOR ALMOST-PERIODIC 2ND-ORDER SYSTEMS

In this paper we prove the existence of at least one homoclinic soluti on for a second order Lagrangian system, where the potential is an alm ost periodic function of time. This result generalizes existence theor ems known to hold when the dependence on time of the potential is peri odic. The method is of a variational nature, solutions being found as critical points of a suitable functional. The absence of a group of sy mmetries for which the functional is invariant (as in the case of peri odic potentials) is replaced by the study of problems ''at infinity'' and a suitable use of a property introduced by E. Sere.